1. No category

GEOMETRIC SEQUENCES AND SERIES
PreCalculus Notes Section 9.4 (continued)
GEOMETRIC SEQUENCE: A pattern where the ___________ of consecutive terms is always the same.
ex: 2, 6, 18, 54, 162, …
If you divide the second term by the first, the third term by the second, and so on, what do you get? ______
3 is called the ______________________, denoted r.
𝑟=
In general terms,
Also, generally speaking, a geometric sequence can be written as: {𝑎1 , 𝑎1 𝑟, 𝑎1 𝑟 2 , 𝑎1 𝑟 3 , … }
Ex 1: Show that {𝒂𝒏 } = {𝟐−𝒏 } is geometric.
“Show” means to prove that it is geometric for ANY set of n’s that you plug in… not just a couple of random numbers
that you decide to try out. So we need to “show” using the above definition for r and use n and show that we get
that r is a constant.
Solution:
𝟏 𝒏
Ex 2: Is{𝒃𝒏 } = {−𝟑 (𝟒) } a geometric sequence?
Solution:
Ex 3: Is {𝒂𝒏 } = {𝟐𝒏} a geometric sequence?
Solution:
FINDING A RULE FOR A GEOMETRIC SEQUENCE:
Formula:
{𝒂𝒏 } =
Ex 4: Find the rule for the sequence listed at the beginning of the notes.
Solution:
Ex 5: Find the 17th term of the geometric sequence 2, 2/3, 2/9, 2/27, …
Use the formula with n = 17. Figure out what r is by dividing second term by first: r = _______
Solution:
Ex 6: Given a1 = 2, r = 4, find the 15th term.
Plug the given info into the formula and use n = 15.
Solution:
ADDING THE FIRST N TERMS OF A GEOMETRIC SEQUENCE:
𝑆𝑛 =
Formula:
𝟏
𝟐
𝟐𝟐
𝟐𝟑
𝟐𝒏−𝟏
Ex 7: Find the general sum of 𝟒 + 𝟒 + 𝟒 + 𝟒 + ⋯ + 𝟒 .
In order to use the formula, we need r, so divide the second term by the first. r = ______
Solution:
𝟏
𝟐
𝟐𝟐
𝟐𝟑
𝟐𝟏𝟒
Ex 8: Find the sum of 𝟒 + 𝟒 + 𝟒 + 𝟒 + ⋯ + 𝟒 .
Since we already found the general equation for the sum of this sequence, we just need to plug in the value of n that
created the last term listed. n =15.
Solution:
OR in calculator: sum(seq(
INFINITE SERIES:
An infinite geometric series is the SUM of a geometric sequence that gets infinitely smaller (so 0 < r < 1).
Notation:
∞
∑ 𝑎1 𝑟 𝑛−1
𝑘=1
1−𝑟 𝑛
If we use the sum formula listed above, 𝑆𝑛 = 𝑎1 ( 1−𝑟 ) and consider that 0 < r < 1 and look at what happens as n
gets closer to infinity, we get that:
geometric series =
𝟏
𝟏
Ex 9: Find 𝟏 + 𝟑 + 𝟗 + …
We need to use the formula above, so we need r. Divide the second term by the first. r = ______
Solution:
This should seem like a strange concept – the idea that we are adding and getting a solution to something that never
ends… But when happens to the terms toward the “end” of the series? They end up getting so small that they are
no longer really affecting the sum when you add them to the previous terms. What is this concept similar to?
Ex 10: Find 8 + 4 + 2 + …
Find r by dividing the second term by the first. r = ______
Solution:
Ex 11:
∞
2 𝑘−1
∑ 2( )
3
𝑘=1
From this notation, we get that a1 = ______ and r = ______. Plug them into the formula for a series.
Solution: